# How do you graph the polar equation r=sin4theta?

Jul 18, 2018

See the graph and the explanation.

#### Explanation:

If p an q are co-prime positive integers, the number of petals

created by either $r = \sin \left(\frac{p}{q}\right) \theta \mathmr{and} r = \cos \left(\frac{p}{q}\right) \theta$ is p.

Here. $r = \sin 4 \theta$ and p = 4. Use

$r = \sqrt{{x}^{2} + {y}^{2}} , \left(x , y\right) = r \left(\cos \theta , \sin \theta\right)$.and

$\sin 4 \theta = 4 {C}_{1} {\cos}^{3} \theta \sin \theta - 4 {C}_{3} \cos \theta$

$= 4 \sin \theta \cos \theta \left({\cos}^{2} \theta - {\sin}^{2} \theta\right)$

and obtain the Cartesian form of $r = \sin 4 \theta$ as

${\left({x}^{2} + {y}^{2}\right)}^{2.5} - 4 x y \left({x}^{2} - {y}^{2}\right) = 0$.

The Socratic graph is immediate.

graph{ ( x^2 + y^2 )^2.5 - 4 x y ( x^2 - y^2 ) = 0[-2.2 2.2 -1.1 1.1] }

The number of petals of $r = \sin \left(\frac{4}{3}\right) \theta$ is also 4#. See the idiosyncratic petals.

graph{3xy(x^2-y^2)-4(x^2+y^2)^2sqrt(1-(x^2+y^2))(1-2(x^2+y^2))=0[-4 4 -2 2]}.